A train travels with a speed of 60 km `h^(-1)`' from station A to station B and then comes back with a speed 80 km `h^(-1)`' from station B to station A. Find : (i) the average speed, and (ii) the average velocity of train.

Explanation

To solve the problem, we need to calculate both the average speed and the average velocity of the train. Let's break it down step by step.

### Step 1: Define the distance
Let the distance between station A and station B be \( x \) kilometers.

### Step 2: Calculate total distance traveled
The train travels from A to B and then back from B to A. Therefore, the total distance traveled by the train is:
\[ \text{Total Distance} = x + x = 2x \text{ kilometers} \]

### Step 3: Calculate time taken for each leg of the journey
1. **Time taken to travel from A to B** at a speed of 60 km/h:
\[ t_1 = \frac{x}{60} \text{ hours} \]

2. **Time taken to travel from B to A** at a speed of 80 km/h:
\[ t_2 = \frac{x}{80} \text{ hours} \]

### Step 4: Calculate total time taken
The total time taken for the entire journey is the sum of the time taken for each leg:
\[ \text{Total Time} = t_1 + t_2 = \frac{x}{60} + \frac{x}{80} \]

To add these fractions, we need a common denominator. The least common multiple of 60 and 80 is 240.
\[ t_1 = \frac{x}{60} = \frac{4x}{240}, \quad t_2 = \frac{x}{80} = \frac{3x}{240} \]
Now, adding these:
\[ \text{Total Time} = \frac{4x}{240} + \frac{3x}{240} = \frac{7x}{240} \text{ hours} \]

### Step 5: Calculate average speed
The average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2x}{\frac{7x}{240}} = 2x \times \frac{240}{7x} = \frac{480}{7} \approx 68.57 \text{ km/h} \]

### Step 6: Calculate average velocity
Average velocity is defined as:
\[ \text{Average Velocity} = \frac{\text{Displacement}}{\text{Total Time}} \]
Since the train returns to its starting point (station A), the total displacement is zero:
\[ \text{Displacement} = 0 \text{ km} \]
Thus, the average velocity is:
\[ \text{Average Velocity} = \frac{0}{\text{Total Time}} = 0 \text{ km/h} \]

### Final Answers
1. Average Speed: \( 68.57 \text{ km/h} \)
2. Average Velocity: \( 0 \text{ km/h} \)

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